On a Vizing-like conjecture for direct product graphs.

Department of Mathematics, PEF, University of Maribor, Koroška cesta 160, 2000 Maribor, Slovenia
Discrete Mathematics (Impact Factor: 0.58). 01/1996; 156:243-246. DOI: 10.1016/0012-365X(96)00032-5
Source: DBLP

ABSTRACT Let fl(G) be the domination number of a graph G, and let G \Theta H be thedirect product of graphs G and H . It is shown that for any k 0 there existsa graph G such that fl(G \Theta G) fl(G)

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    ABSTRACT: An upper bound for the domination number of the direct product of graphs is proved. It in particular implies that for any graphs G and H, γ(G×H)⩽3γ(G)γ(H). Graphs with arbitrarily large domination numbers are constructed for which this bound is attained. Concerning the upper domination number we prove that Γ(G×H)⩾Γ(G)Γ(H), thus confirming a conjecture from [R. Nowakowski, D.F. Rall, Associative graph products and their independence, domination and coloring numbers, Discuss. Math. Graph Theory 16 (1996) 53–79]. Finally, for paired-domination of direct products we prove that γpr(G×H)⩽γpr(G)γpr(H) for arbitrary graphs G and H, and also present some infinite families of graphs that attain this bound.
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